Respuesta :
Answer:
The value of a+b is 4.
Step-by-step explanation:
The given function is
[tex]\[f(x) = \left\{ \begin{array}{cl} 9 - 2x & \text{if } x \le 3, \\ ax + b & \text{if } x > 3. \end{array} \right.\][/tex]
It is given that for some constants a and b the function f has the property that f(f(x))=x for all x.
For x≤3,
[tex]f(x)=9-2x[/tex]
For x>3,
[tex]f(x)=ax+b[/tex]
At x=0,
[tex]f(0)=9-2(0)=9[/tex]
[tex]f(f(0))=f(9)\Rightarrow a(9)+b=9a+b[/tex]
Using property f(f(x))=x,
[tex]f(f(0))=0[/tex]
[tex]9a+b=0[/tex] .... (1)
At x=1,
[tex]f(1)=9-2(1)=7[/tex]
[tex]f(f(1))=f(7)\Rightarrow a(7)+b=7a+b[/tex]
Using property f(f(x))=x,
[tex]f(f(1))=1[/tex]
[tex]7a+b=1[/tex] .... (2)
Subtract equation (2) from equation (1).
[tex]9a+b-(7a+b)=0-1[/tex]
[tex]2a=-1[/tex]
Divide both sides by 2.
[tex]a=-\frac{1}{2}[/tex]
Substitute this value in equation (1).
[tex]9(-\frac{1}{2})+b=0[/tex]
[tex]b=\frac{9}{2}[/tex]
The value of a is [tex]-\frac{1}{2}[/tex] and value of b is [tex]\frac{9}{2}[/tex]. The value of a+b is
[tex]a+b=-\frac{1}{2}+\frac{9}{2}[/tex]
[tex]a+b=4[/tex]
Therefore the value of a+b is 4.
